Euclidean vector space/R^3/Proper isometry/Fact/Proof

Proof

We consider the characteristic polynomial of ${}\varphi$, that is,

${}P(\lambda )=\det {\left(\lambda E_{3}-\varphi \right)}\,.$

This is a normed real polynomial of degree three. For ${}\lambda =0$, we have

${}P(0)=\det {\left(-\varphi \right)}=-\det \varphi =-1\,.$

As the polynomial ${}P(\lambda )\to \infty$ as ${}\lambda \to \infty$, there must be a positive ${}\lambda$ that is a zero of ${}P$. Due to fact, we must have ${}\lambda =1$.

To fact